Inspired from discussions in [1]}, it is sufficient to prove existence of \(\hat{\alpha }>0\) , \(\beta >0\)  and \(C>0\)  such that \(\beta < \hat{\alpha }\)  and:
\(& \nonumber \int e^{\hat{\alpha } \Vert y\Vert ^{2}}\mathcal {P}(x,dy) \le Ce^{\beta \Vert x\Vert ^{2}}, \hspace{10.0pt} \text{for all }~x \in \mathcal {X}. \hspace{5.0pt} \text{i.e.,}~  \\ & \int e^{\hat{\alpha } \Vert y\Vert ^{2}}\mathcal {P}(x,dy) \le Ce^{(\beta -\hat{\alpha })\Vert x\Vert ^{2}}e^{\hat{\alpha }\Vert x\Vert ^{2}}\) 
